This paper is connected with recent results of Duren and Pfaltzgraff (J. Anal. Math., 78, 205-218 (1999)). We consider the problem on the distortion of the hyperbolic Robin capacity δ h (A, Ω) of the boundary set A ⊂ ∂Ω under a conformal mapping of a domain Ω ⊂ U into the unit disk U. It is shown that, for sets consisting of a finite number of boundary arcs or complete boundary components, the inequality cap hf(A) ≥ δh (A,Ω) (*) is sharp in the class of conformal mappings f: Ω → U such that f(∂U) = ∂U. Here cap h f(A) is the hyperbolic capacity of a compact set f(A) ⊂ U. We give some examples demonstrating properties of functions which realize the case of equality in relation (*).